Supplement 3.2: Polarisation of Electromagnetic Waves: Jones Vectors and Jones Matrices (2/2)

Optical interactions as 2×2 matrix: Jones Matrices

Following an optical interaction we mark the field components as dotted quantities. They are a linear combination of the primary components:

Written as a matrix:

The amplitude-transformation matrix $A$ is the Jones Matrix.

It is the aim to find a Jones Matrix for every type of optical components which changes the polarisation of light. Actually, the optical effect of those components depends on their orientation in relation to the light beam: A linear polariser for example influences polarised light depending on its orientation in completely different ways. This applies to the same extent to the splitting of light beams when hitting a refractive surface (glass, water...).

This may be found by formulating a single Jones Matrix for a defined orientation of all components or optical interactions that influence the polarisation. The matrix for other orientations may be set up by adjusting the polarised light by rotating the y and z coordinates to the position in which it should interact with the component and in the following bringing it back to the initial position.

A rotation of the y and z coordinates by an angle $\alpha$ into another y' and z' coordinate system as shown in the following graphics ...

... is calculated as follows:

 

The coefficients of the system of equations are the elements of the rotation matrix:

If we name the amplitude-transformation matrices of components with a defined orientation as $A$ and the same components at an angle $\alpha$ with $A(\alpha )$, then it becomes:

Examples of Jones Matrices are shown in the following table. They hold for ideal properties, that means absorption- and reflection-free components and perfectly plane surfaces.

Jones Matrix	Component or interaction
$\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)$	Linear polariser transmitting along the y axis
$\left(\begin{array}{cc}{\text{e}}^{-i\phi /2}& 0\\ 0& {\text{e}}^{i\phi /2}\end{array}\right)$	Retarder with a phase difference $\phi$ of the wave parts, fast axis in y direction
$\left(\begin{array}{cc}{\text{e}}^{-i\pi /4}& 0\\ 0& {\text{e}}^{i\pi /4}\end{array}\right)$	$\lambda /4$ retarder ($\lambda /4$ waveplate) with $\phi =\pi /2$, fast axis in y direction
$\left(\begin{array}{cc}{\text{e}}^{-i\pi /2}& 0\\ 0& {\text{e}}^{i\pi /2}\end{array}\right)$	$\lambda /2$ retarder ($\lambda /2$ waveplate) with $\phi =\pi$, fast axis in y direction
$\left(\begin{array}{cc}{t}_{\parallel }& 0\\ 0& t_{\perp }\end{array}\right)$	Fresnel refraction at a dielectric material with an angle of incidence ${\delta }_i$ and angle of refraction ${\delta }_t$, the y axis is in the plane of incidence and the z axis is perpendicular to the plane of incidence, and with the Fresnel amplitude coefficients for refraction: $\begin{array}{l}{t}_{\parallel }=2sin{\delta }_{t}cos{\delta }_i/\left(sin({\delta }_i+{\delta }_t)cos({\delta }_i-{\delta }_t)\right)\\ t_{\perp }=2sin{\delta }_{t}cos{\delta }_i/sin({\delta }_i+{\delta }_t)\end{array}$
$\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)\cdot \left(\begin{array}{cc}{r}_{\parallel }& 0\\ 0& r_{\perp }\end{array}\right)$	Fresnel reflection at a dielectric material with an angle of incidence ${\delta }_i$ and angle of refraction ${\delta }_t$, the y axis is in the plane of incidence and the z axis is perpendicular to the plane of incidence, and with the Fresnel amplitude coefficients for reflection: $\begin{array}{l}{r}_{\parallel }=-tan\left({\delta }_i-{\delta }_t\right)/tan\left({\delta }_i+{\delta }_t\right)\\ r_{\perp }=-sin\left({\delta }_i-{\delta }_t\right)/sin\left({\delta }_i+{\delta }_t\right)\end{array}$ The left matrix results from the directional change of the light propagation due to reflection.
$\left(\begin{array}{cc}cos\alpha & -sin\alpha \\ sin\alpha & cos\alpha \end{array}\right)$	Rotation by the angle $\alpha$

The effect of several components in a row would be calculated from the product of the respective matrices. For n components:

where $A_1$ is the first and $A_n$ is the last transmitted component in a row. The matrices are not commutative (inverting the components would be of consequences as well).